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This work studies a class of non-smooth decentralized multi-agentoptimization problems where the agents aim at minimizing a sum of localstrongly-convex smooth components plus a common non-smooth term. We propose ageneral primal-dual algorithmic framework that unifies many existingstate-of-the-art algorithms. We establish linear convergence of the proposedmethod to the exact solution in the presence of the non-smooth term. Moreover,for the more general class of problems with agent specific non-smooth terms, weshow that linear convergence cannot be achieved (in the worst case) for theclass of algorithms that uses the gradients and the proximal mappings of thesmooth and non-smooth parts, respectively. We further provide a numericalcounterexample that shows how some state-of-the-art algorithms fail to convergelinearly for strongly-convex objectives and different local non-smooth terms.

Decentralized Proximal Gradient Algorithms with Linear Convergence Rates